Identification of criticality in neuronal avalanches: I. A theoretical investigation of the non-driven case

TL;DR

This paper investigates a theoretical model of a neural network at criticality, showing that scale-free avalanche distributions are not necessarily power laws and emphasizing the importance of multiple markers for identifying criticality.

Contribution

It provides an exact analysis of avalanche size distributions in a finite-size neural model, highlighting limitations of power law fitting for criticality detection.

Findings

Avalanche size distribution shows scale-free behavior but is not a power law.

Susceptibility diverges at the critical point, serving as a marker.

Power laws may underlie other system observables, aiding experimental detection.

Abstract

In this paper we study a simple model of a purely excitatory neural network that, by construction, operates at a critical point. This model allows us to consider various markers of criticality and illustrate how they should perform in a finite-size system. By calculating the exact distribution of avalanche sizes we are able to show that, over a limited range of avalanche sizes which we precisely identify, the distribution has scale free properties but is not a power law. This suggests that it would be inappropriate to dismiss a system as not being critical purely based on an inability to rigorously fit a power law distribution as has been recently advocated. In assessing whether a system, especially a finite-size one, is critical it is thus important to consider other possible markers. We illustrate one of these by showing the divergence of susceptibility as the critical point of the system is approached. Finally, we provide evidence that power laws may underlie other observables of the system, that may be more amenable to robust experimental assessment.